Higher-dimensional topological phases play a key role in understanding the lower-dimensional topological phases and the related topological responses through a dimensional reduction procedure. In this work, we present a Dirac-type model of four-dimensional ${\mathbb{Z}}_2$ topological insulator (TI) protected by $\mathcal{CP}$ symmetry, whose 3D boundary supports an odd number of Dirac cones. A specific perturbation splits each bulk massive Dirac cone into two valleys separated in energy-momentum space with opposite second Chern numbers, in which the 3D boundary modes become a nodal sphere or a Weyl semimetallic phase. By introducing the electromagnetic (EM) and pseudo-EM fields, exotic topological responses of our 4D system are revealed, which are found to be described by the $(4+1)\mathrm{D}$ mixed Chern-Simons theories in the low-energy regime. Notably, several topological phase transitions occur from a $\mathcal{CP}$-broken ${\mathbb{Z}}_2$ TI to a $\mathbb{Z}$ TI when the bulk gap closes by giving rise to exotic double-nodal-line or nodal-hyper-torus gapless phases. Finally, we propose to probe experimentally these topological effects in cold atoms.

• Received 7 April 2022
• Revised 29 September 2022
• Accepted 19 October 2022

DOI:https://doi.org/10.1103/PhysRevLett.129.196602